Rates and transport

Rates and transport

Thermodynamics says whether a step can run. Rates say how long it takes. extensions/chemistry/kinetics.mojo and transport.mojo turn measured rate data into times, depths and temperature bounds.

This page is a reference. For the rest of the library, see Process chemistry.

Arrhenius rates

An Arrhenius rate constant is k = A exp(-Ea / R T), first order, in 1/s.

• rate(temperature) gives k.
• temperature_for(rate) gives the temperature where k reaches a rate.
• half_life(temperature) gives ln 2 / k.

RateAtLeast is a Constraint. It holds where k reaches a minimum. A kinetic bound that practice states as a temperature, such as "above 900 degC", can then come from measured rate data instead.

from extensions.chemistry.kinetics import Arrhenius, RateAtLeast
from extensions.chemistry.quantities import KILOJOULE_PER_MOLE, MolarEnergy64
from units.si import Frequency64

var law = Arrhenius(Frequency64(1e10), MolarEnergy64(150, KILOJOULE_PER_MOLE))
var need = RateAtLeast(law, Frequency64(1e-3))   # add with met_intervals

Gas and solid: the shrinking core

ShrinkingCore is the model of Levenspiel ("Chemical Reaction Engineering", chapter 25) for a gas that reacts with a solid sphere. Three resistances act in series: the gas film, the layer of product ash, and the reaction at the core's surface. The time to convert a share X of the sphere is:

t = tau_film X
  + tau_ash (1 - 3 (1 - X)^(2/3) + 2 (1 - X))
  + tau_reaction (1 - (1 - X)^(1/3))

ShrinkingCore.from_properties computes the three times from the properties of the sphere and the gas. These are the radius, the solid's molar density, the gas concentration and the stoichiometry. They also include the film coefficient, the ash diffusivity and the surface rate constant. time_to(conversion) gives the time, and conversion_at(time) inverts it by bisection. The largest of the three times shows which resistance controls the step.

Evaporation

evaporation_flux is the Hertz-Knudsen-Langmuir rate at which a surface loses mass into a vacuum or a thin gas:

J = alpha (p_sat - p) sqrt(M / (2 pi R T))

vapor_pressure(condensed, vapor, temperature) finds p_sat from the data. evaporation_time turns a flux into the time to remove a mass from an area. The vacuum distillation of a Kroll sponge is an example: it boils the leftover Mg and MgCl2 away.

Diffusion into a surface

Diffusivity is an Arrhenius diffusion coefficient, D = D0 exp(-Q / R T). case_depth and time_to_depth solve Fick's second law for a semi-infinite solid with a fixed surface concentration:

(C - C0) / (Cs - C0) = erfc(x / (2 sqrt(D t)))

This is the case depth of carburizing and nitriding.

Heating a part

heating_time is the lumped-capacitance time to heat a body in a furnace. It is valid where the Biot number h L / k is below about 0.1. biot gives that number, and heating_time raises an error above the limit.

Checks

Quantity	Computed	Reference
Carburizing time to 0.80 wt% C at 0.5 mm, 950 degC	7.05 h	About 7.1 h, Callister
Mg vapor pressure at 971 K	Within 10 % of 1 kPa	1 kPa, CRC Handbook
Shrinking core, one resistance at a time	Equal	Levenspiel's closed forms

Limits

A rate constant comes from a measurement on the real material. The library does not predict one. The shrinking-core model takes spheres of one size, and it does not model the heat of the reaction.